DCM for fMRI:
Theory & Practice
Peter Zeidman and Laura Madeley
Overview
• Theory
– Why DCM?
– What DCM does
– The State Equation
• Application
– Planning DCM studies
– Hypotheses
– How to complete in SPM
Brains as Systems
Source: http://public.kitware.com/ImageVote/images/17/
Brains as Systems
Blobs on Brains do not a network make
Taxonomy of Connectivity
1. Structural connectivity – the physical structure of the brain
2. Functional connectivity – the likelihood that 2 neuronal populations share associated
activity
3. Effective connectivity – a union between structural and functional connectivity.
The Challenge
How to infer causality
especially with poor temporal resolution?
DCM Overview
1. Create a neural model to represent our
hypothesis
2. Convolve it with a haemodynamic model to
predict real signal from the scanner
3. Compare models in terms of model fit and
complexity
The Neural Model
Recipe
z4
Z - Regions
z2
z3
z1
The Neural Model
Recipe
z4
z2
z3
z1
Z - Regions
A - Average
Connections
The Neural Model
Attention
Recipe
z4
z2
z3
z1
Z - Regions
A - Average
Connections
B - Modulatory
Inputs
The Neural Model
Attention
Recipe
z4
z2
z3
z1
Z - Regions
A - Average
Connections
B - Modulatory
Inputs
C - External
Inputs
DCM Overview
Neural Model
Haemodynamic Model
4
2
3
1
e.g. region 2
x
=
DCM Overview
=
Region 2 Timeseries
Dynamical Systems
• Less exciting example: interest
• My bank gives me 3% interest on savings
• How will it grow?
Dynamical Systems
The Lorentz Attractor – A Spatial Dynamical System
Dynamical Systems
We can represent a network as a matrix
From
z1
=
To
z2
z3
(The A-Matrix)
Simplified Network Model
z1
z2
z3
z (t )  Az (t  1)
Simplified Network Model
z (t )  A * z (t  1)
z (1)  A * z (0)
0 1 0 1 




 1 0 0 * 0
0 1 0 0
0 


 1 
0
Simplified Network Model
z1
z2
z3
z1
z2
z3
Simplified Network Model
z (2)  A * z (1)
0 1 0 0




 1 0 0 * 1 
0 1 0 0
1 


 0 
1 
Simplified Network Model
z1
z2
z3
z1
z2
z3
Simplified Network Model
z (t  1)  Az (t )
Modulatory Inputs
External Inputs
{
{
z  ( A   u j B ) z  Cu
j
The DCM State Equation
Summary So Far
• The Brain is a Dynamical System
• The simple DCM “forward model” predicts
neural activity
• The software (SPM) combines it the
haemodynamic model, to predict the fMRI
timeseries
• We compare models to choose the one best
matching the real fMRI timeseries
DCM Motivation
Dynamic Causal Modelling (DCM) needed due to a simple problem:
• Cognitive neuroscientists want to talk about activation at the
level of neuronal systems to hypothesize about cognitive
processes
• Imaging techniques do not generate data at this level, but give
output relating to non-linear correlates e.g. fMRI
haemodynamic response (BOLD signal)
• It would be useful to be able to talk about causality in neuronal
populations, since we know that signals propagate from some
input through a system
• DCM attempts to tackle these problems
DCM History
• Introduced in 2002 for fMRI data (Friston, 2002)
• DCM is a generic approach for inferring hidden (unobserved)
neuronal states from measured brain activity.
• The mathematical basis and implementation of DCM for fMRI
have since been refined and extended repeatedly.
• DCMs have also been implemented for a range of measurement
techniques other than fMRI, including EEG, MEG (to be
presented next week), and LFPs obtained from invasive
recordings in humans or animals.
What do we want from fMRI studies?
Different Questions re brain function:
1. Functional specialisation
– Where is a stimulus processed?
– What are regionally specific effects?
• Normal SPM analysis (GLM)
PMd
2. Functional integration
PMv
SMA
– How does the system work?
M1
– What are inter-regional effects?
– How do components of that system interact?
PMd
SMA
PMv
M1
DCM: Basic idea
• DCM allows you to model brain activity at the neuronal
level (not directly accessible in fMRI) using a bilinear state
equation. This takes into account the anatomical
architecture of the system and the interactions within that
architecture under different conditions of stimulus context
• The modelled neuronal dynamics (z) are transformed into
area-specific BOLD signals(y) by a hemodynamic forward
model (more later)
• The aim of DCM is to estimate parameters at the neuronal
level so that the modelled BOLD signals are most similar to
the experimentally measured BOLD signals
DCM: Methods
• Rules of good practice
– 10 Simple Rules for DCM (2010). Stephan et al.
NeuroImage, 52
– Experimental design
– Model Specification
– Decision tree
• DCM in SPM
– Steps within SPM
– Example
• attention to motion in the visual system (Büchel & Friston
1997, Cereb. Cortex, Büchel et al. 1998, Brain)
– http://www.fil.ion.ucl.ac.uk/spm/data/
DCM: Rules of good practice
• Experimental design
– DCM is dependent on experimental perturbations
– Experimental conditions enter the model as inputs that either drive the
local responses or change connection strengths.
– If there is no evidence for an experimental effect (no activation
detected by a GLM) → inclusion of this region in a DCM is not
meaningful.
– Use the same optimization strategies for design and data acquisition that
apply to conventional GLM of brain activity:
– preferably multi-factorial (e.g. 2 x 2)
– one factor that varies the driving (sensory) input (eg static/
moving)
– one factor that varies the contextual input (eg attention / no
attention)
DCM: Rules of good practice II
Defining the model
Model selection = determine which model, from a set of plausible
alternatives, is most useful i.e., represents the best balance between
accuracy and complexity.
So....
•
define models that are plausible & based on neuroimaging, electrophysiology,
TMS, studies
• Use anatomical information and computational models to refine DCM
• model definition should be as transparent and systematic as possible
• How many plausible model alternatives exist?
• For small systems it is possible to investigate all possible connectivity
architectures.
• With increasing number of regions and inputs, evaluating all possible models
becomes practically impossible very rapidly.
• The map is not the territory.....
• Models are caricatures of complex phenomena, enabling testing of
underlying mechanisms.
Fig. 1. This schematic summarizes the typical sequence of analysis in DCM, depending on the question of interest. Abbreviations: FFX=fixed
effects, RFX=random effects, BMS=Bayesian model selection, BPA=Bayesian parameter averaging, BMA=Bayesian model averaging,
ANOVA=analysis of variance.
10 Simple Rules for DCM (2010). Stephan et al. NeuroImage 52.
Attention to motion in the visual system
Stimuli 250 radially moving dots at 4.7 degrees/s
Pre-Scanning
5 x 30s trials with 5 speed changes (reducing to 1%)
Task - detect change in radial velocity
Scanning (no speed changes)
6 normal subjects, 4 x 100 scan sessions;
each session comprising 10 scans of 4 different
conditions
F A F N F A F N S .................
PPC
V3A
V5+
F - fixation point only
A - motion stimuli with attention (detect changes)
N - motion stimuli without attention
S - no motion
Attention – No attention
Büchel & Friston 1997, Cereb. Cortex
Büchel et al. 1998, Brain
Practical steps of a DCM study - I
1. Definition of the model
Model 2:
attentional modulation
• Structure: which areas,
of V1→V5
connections and inputs?
Photic
PPC
0.85
0.70
• Which parameters represent
0.84
1.36
V1
my hypothesis?
-0.02
0.57
• What are the alternative
V5
0.23
Motion
models to test?
Attention
2. Defining criteria for inference:
• single-subject analysis: stat. threshold? contrast?
• group analysis: which 2nd-level model?
3. Conventional SPM analysis (subject-specific)
• DCMs fitted separately for each session
→
consider concatenation of sessions or
adequate 2nd level analysis
Practical steps of a DCM study - II
4. Extraction of time series, e.g. via VOI tool in SPM
• n.b anatomical & functional standardisation important for
group analyses
5. Possible definition of a new design matrix, if the “normal”
design matrix does not represent the inputs appropriately.
• DCM only reads timing information of each input from the
design matrix, no parameter estimation necessary
6. Definition of model
• via DCM-GUI or directly
in MATLAB
Practical steps of a DCM study - III
7.
DCM parameter estimation
• Remember models with many regions & scans can crash
MATLAB!
8.
Model comparison and selection:
• Which of all models considered is the optimal one?
 Bayesian model selection tool
9.
Testing the hypothesis
Statistical test on
the relevant parameters
of the optimal model
Specify design matrix & SPM analysis
Contextual factor
No
attent
Attent.
No motion/
no attention
No motion/
attention
moving
Motion /
no attention
Motion /
attention
Normal SPM regressors
DCM analysis regressors
•
-Vision (photic)
•
-motion
•
-attention
Attention
static
Photic
Sensory input factor
Motion
Experimental design
Extraction of time series (VOIs definition)
1.
DCM for a single subject analysis (i.e. no 2ndlevel analysis intended):
determine representative co-ordinates for each
brain region from the appropriate contrast (e.g.
V1 from “photic” contrast )
2.
Subject specific DCM, but results will eventually
be entered into a 2nd-level analysis:
determine group maximum for the area of
interest (e.g. from RFX analysis) in the
appropriate contrast
in each subject, jump to local maximum nearest
to the group maximum, using the same contrast
and a liberal threshold (p<0.05, uncorrected)
VOIs definition: V5
Contrast
Name
Co-ordinates
Definition of DCM
name
DCM button
In order!
In Order!!
Output
Average connectivity (A)
Modulation of connections (B)
Photic
Motion
Motion
Attention
Attention
Input (C)
Comparison of two simple models
Model 2:
attentional modulation
of V1→V5
Model 1:
attentional modulation
of PPC→V5
Photic
Attention
Photic
PPC
0.85
0.55
0.86
0.70
PPC
0.75
1.42
0.84
1.36
V1
0.89
V1
0.57
-0.02
V5
0.56
Motion
Motion
-0.02
V5
0.23
Attention
Models comparison and selection
Model 2
better than
model 1
SPM: Issues
What you cannot do with BMS
• A DCM is defined for a specific data set.
Therefore BMS cannot be applied to models
that are fitted to different data
• Cannot compare models with different
numbers of regions, because changing the
regions changes the data
• Maximum of 8 regions with SPM8
• Designed for sensory-driven studies
So, DCM….
• enables one to infer hidden neuronal processes from fMRI data
• tries to model the same phenomena as a GLM
– explaining experimentally controlled variance in local responses
– based on connectivity and its modulation
• allows one to test mechanistic hypotheses about observed effects
• is informed by anatomical and physiological principles.
• uses a Bayesian framework to estimate model parameters
• is a generic approach to modelling experimentally perturbed dynamic
systems.
• provides an observation model for neuroimaging data, e.g. fMRI, M/EEG
• Should be planned for early in the experimental process
References
• The first DCM paper: Dynamic Causal Modelling (2003). Friston et al. NeuroImage
19:1273-1302.
• Physiological validation of DCM for fMRI: Identifying neural drivers with functional
MRI: an electrophysiological validation (2008). David et al. PLoS Biol. 6 2683–2697
• Hemodynamic model: Comparing hemodynamic models with DCM (2007). Stephan
et al. NeuroImage 38:387-401
• Nonlinear DCMs:Nonlinear Dynamic Causal Models for FMRI (2008). Stephan et al.
NeuroImage 42:649-662
• Two-state model: Dynamic causal modelling for fMRI: A two-state model (2008).
Marreiros et al. NeuroImage 39:269-278
• Group Bayesian model comparison: Bayesian model selection for group studies
(2009). Stephan et al. NeuroImage 46:1004-10174
• 10 Simple Rules for DCM (2010). Stephan et al. NeuroImage 52.
• Dynamic Causal Modelling: a critical review of the biophysical and statistical
foundations. Daunizeau et al. Neuroimage (2010), in press
• SPM Manual, SMP courses slides, last years presentations.
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